periodicity????????????

Question

prove that x+sinx is not a periodic function.......plz tell me the procedure..

Answer 1

If x + sin x were a periodic function, there would exist a constant T > 0 such that

x + sin x = (x + T) + sin (x + T),

ie.

sin x - sin (x + T) = T (*)

for all real x. Setting x = - T yields

sin (-T) = - sin T = T,

and setting x = T yields

sin T + sin 2T = T.

Thus

- sin T = T = sin T + sin 2T,

which we can rewrite as

- 2 sin T = sin 2T = 2 sin T cos T,

and so sin T = 0 or cos T = -1.

Thus, T = kπ for some positive integer k. Substituting this into (*) yields

sin x - sin (x + kπ) = kπ,

which for x = 0 implies

- sin kπ = kπ,

which is only possible for k = 0. Thus T = 0, and we have reached a contradiction. Hence x + sin x is not a periodic function.

Answer 2

If f(x) is periodic there exists a number such as
f(x+P) = f(x)Not depending on x
you know that sin x has period 2pi
x+P+sin(x+P) =? x +sin x
so P+sin(x+P) =? sin(x) so -P= sin(x+P)-sin(x)=2cos(x+P/2)*sin(P/2)
So the value of P in this equation depends on x.

Answer 3

for |x|>2pi, |sinx|<1, so x dominates x+sinx
(x)' = 1 so x is always increasing so can not be periodic.